Fundamental theorem of calculus: continuity and oriented endpoints

Mathematical statement

Let f:R→R be continuous and a∈R. For G(x)=∫_a^x f(t)dt using an oriented definite integral, G is differentiable everywhere on R and G′(x)=f(x). If F:R→R is differentiable with F′=f on R, then ∫_a^b f(t)dt=F(b)−F(a) for arbitrary real endpoints a,b. These full-real sufficient premises do not assert endpoint one-sided derivatives or the same result for arbitrary merely integrable functions.

Representative formula (exact TeX)

D(G,x)

Assumptions

• The integrand is continuous on all R. Exercises restrict it to one through four ascending safe-integer polynomial coefficients, degree at most three; candidate primitives have one through five positive-denominator rational coefficients.
• Oriented integrals permit reversed and equal endpoints. G(a)=0 does not imply G′(a)=0. For full-real affine moving bounds, the derivative is f(u(x))u′(x)−f(l(x))l′(x).
• Check F′=f coefficientwise, not by finite sampling. Failure of a candidate does not reject other possible antiderivatives; general FTC has no attached formal Evidence.

Symbol scope

• D — real derivative at a specified point
• G — accumulating oriented integral of a continuous function
• x — real derivative center

Intuition, definition, and misconceptions

The accumulation value at its anchor is zero, but its derivative is f(a). Equal moving bounds at a point do not force a zero derivative, because their velocities may differ.

The endpoint formula first requires F′=f. Adding a constant does not change the endpoint difference, while reversing endpoints reverses its sign. Exact polynomial and rational checks are not a general formal proof of the theorem.

Common errors include inferring a zero derivative from a zero anchor integral or coincident bounds, or ignoring the antiderivative premise.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real derivatives at a point and finite difference quotients — Retain derivative existence and the two-sided limit.
• Signed polynomial definite integrals on finite intervals — Use oriented endpoints and the continuous integrable slice.
• Point values and continuity of real functions — Continuity on all R is the chosen sufficient premise.
• Polynomial antiderivatives and the choice of constant — Require an eligible primitive with F′=f.
• Differentiating polynomial sums, products, and compositions — Use the chain rule for affine moving bounds.

Verification boundary

This theorem candidate has explicit sufficient premises. Finite polynomial computations can be replayed; general FTC and the Riemann-to-Mathlib interval-integral bridge remain unformalized.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind pinned intervalIntegral.integral_hasDerivAt_right and integral_eq_sub_of_hasDerivAt, with continuity/integrability, Riemann agreement, and endpoint-orientation type bridges.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.